I saw a demonstration in which a lightbulb placed in a microwave
oven glowed. Why did it?

John Campbell, who teaches physics at the University of Canterbury, responded.

Before anyone tries this, keep in mind that a microwave oven uses up
to 600 watts of power whereas a standard lightbulb takes only 100 watts
to make it incandescent, so the bulb can be destroyed doing this.
I use a bulb with a broken filament, and also immerse the electrical
contact end in a bowl of cooling water.

Metals contain some electrons which are free to move whenever an
electrical force is applied to them. When an electrical current is forced
through a metal wire the travelling electrons jostle the atoms of the
metal, so heating up the wire.

The microwave oven works by generating very high frequency microwaves
which oscillate at nearly ten thousand million times every second.

To prevent ourselves being cooked too, the microwaves are sprayed into a
metal box. There are no electrical forces outside the box. There are also
none at the inside surface of the metal box because the electrons move to
cancel the force. (I show this in classes by putting grated cheese on
toast against the metal wall. The cheese next to the wall doesn't melt.)

The metal walls reflect electric waves so there are places inside the oven
where the two waves reinforce, giving a maximum of electrical force, and
other places where the two waves cancel, giving a minimum of electrical
force. These"hot" and "cool" spots explain why food must be rotated
inside a microwave oven.

Hence the electric force along the filament isn't uniform so currents
flow in the filament to try to cancel out the forces.
Your light bulb will wax and wane as it rotates.

This heating of metals explains the advice to not use dishes with gold
designs in microwave ovens. The thin gold foil can be heated to evaporation.
A gold ring will also cause a ring of heating, and the stress can crack
the dish.